Solid Conducting sphere inside a conducting spherical shell

Consider a solid conducting sphere with a radius 0.7 cm and charge −7.3 pC on it. There is a conducting spherical shell concentric to the sphere. The shell has an inner radius 2.8 cm(with 2.8 cm > 0.7 cm) and outer radius 5 cm and a net charge 27.9 pC on the shell. Denote the charge on the inner surface of the shell by Q′2 and that on the outer surface of the shell by Q′′2. Find the charge Q′′2. Answer in units of pC

2. Relevant equations

I got this problem wrong and don't know why.
Shouldn't Q''2 be 0 since the sphere is negative on the surface and attracting the positive charges on the inner shell. The shell should be stable since its a conductor and its Electric field inside is=0. Therefore, the Charge outside the shell is 0.

3. The attempt at a solution
Conceptually I thought
inside the shell should be the opposite of the charge of the sphere. Since the sphere is negative, then the charge Q'2 inside surface of the shell is same charge but positive. So Q inside shell + Q'2 inside surface of shell = Q''2 outside shell (of whole sphere). For the conducting sphere to be in equilibrium Q''2 should just be 0.

Consider a solid conducting sphere with a radius 0.7 cm and charge −7.3 pC on it. There is a conducting spherical shell concentric to the sphere. The shell has an inner radius 2.8 cm(with 2.8 cm > 0.7 cm) and outer radius 5 cm and a net charge 27.9 pC on the shell. Denote the charge on the inner surface of the shell by Q′2 and that on the outer surface of the shell by Q′′2. Find the charge Q′′2. Answer in units of pC

2. Relevant equations

I got this problem wrong and don't know why.
Shouldn't Q''2 be 0 since the sphere is negative on the surface and attracting the positive charges on the inner shell. The shell should be stable since its a conductor and its Electric field inside is=0. Therefore, the Charge outside the shell is 0.

3. The attempt at a solution
Conceptually I thought
inside the shell should be the opposite of the charge of the sphere. Since the sphere is negative, then the charge Q'2 inside surface of the shell is same charge but positive. So Q inside shell + Q'2 inside surface of shell = Q''2 outside shell (of whole sphere). For the conducting sphere to be in equilibrium Q''2 should just be 0.

The essential point in this problem is that the electric field inside a conducting object is zero when the charges are in equilibrium. Therefore the charge on the outer spherical shell arranges itself so that it cancels the electric field due to the inner solid conducting sphere. That means there will be some charge left over and this charge will flow to the outer surface. Remember free charge on a conductor flows to the outer surface. (Gauss's law is used to prove this.)